An analysis of the problem instance is – from the user of point of view – just a container for a set of iterations done for by a user for a selected problem instance. This approach provides a structure for possibly large sets of iterations, and also to assure privacy (the user of the MCAA has access only to iterations they he/she has created).
First iteration is generated automatically (with equal preferences for all criteria se-lected for the corresponding model instance). Each user created iteration starts from a previously done iteration that is selected by the user. Therefore the user implicitly creates a tree of iterations. To create a new iteration the user:
•Selects a parent iteration.
•Selects a MC method (from the set of methods described in Sec. 8), if more then one method is provided for the user.
•Specifies the preferences (Sec. 4).
•After specifying the preferences clicks on the Solve button.
For each selected iteration the user may:9
•Attach a note.
•Examine the chart of alternative characteristics.?
•Examine the chart comparing alternatives by criteria.?
•Browse through the problem instance data.10?
•Users having access to more than one analysis can browse (by clicking on the Return button) the tree composed of: problems, instances, and analyses.
•Browse through the help.?
4 Specification of preferences
4.1 Required for all methods
•ri - relative importance of criteria, specified for each criterion by a button position;
currently 8 buttons (numbered by 0 through 7):
9Each of the actions marked below by the?characters results in opening a new window. Each of these windows (except of the help) is labeled by the a string composed of: problem name, instance name, analysis name, iteration number, and the symbol of the MC method used for this iteration.
10The data are of course common for all iterations, however the access is provided from each iteration.
?0-th button: ignore the criterion;
Note: criteria ”below” (i.e., children, grandchildren, . . . ) ignored criteria are assumed to be also ignored (therefore solvers redefine the specified values of correspondingrii to 0).
?4-th button: average importance;
?buttons 5 through 7: more, much more, vastly more, important than average, respec-tively;
?buttons 3 through 1: less, much less, vastly less, important than average, respectively;
4.1.1 Required for some methods
•impr - selection of criteria that shall be improved and those to be compromised; this is specified for each criterion by a button position; currently 4 buttons (numbered by 0 through 3):
?0th button: allow to compromise (worsen) the criterion value;
?1st button: free the criterion (change in any direction);
?2nd button: stabilize the criterion value (preference for keeping changes small);
?3rd button: improve the criterion value;
4.1.2 Optional or computed from data
•res reservation andasp aspiration values for each criterion
•rfp reference point (one value for each criterion)
5 Representation of preferences in MC solvers
The preferences are specified for each iteration (except of the initial one). First, the user optionally selects for each iteration the method which will be used to find a Pareto solution that fits best his/her preferences. The way the preferences are specified depends on the method so advanced users may experiment with different methods and find the favorite one. Each method uses the associated solver. The methods/solvers differ by the internal representation of user preferences, and the way in which a Pareto solution is selected for specific preferences. However, several elements of solvers are common, and are therefore presented before each method will be outlined with method-specific elements.
All methods use the following six11types of objects and corresponding functions:
•selection of active leaf-criteria
•selection of Pareto alternatives
•wi(ri)andvi(wi)- criteria scaling/weighting;
•IAi(qi) - Individual Achievement functions measuring (for each criterion separately) the satisfaction level corresponding to a value of the criterion;
•AFi(w,v,IA)- Achievement Function measuring (for each criterion) the satisfaction level corresponding to a value of the criterion taking into account relative importance (represented bywi(ri)or/andvi(wi)) of all criteria;
•SF(AF)- Scalarizing Function measuring satisfaction levels for each alternative.
11We present here only a subset of solver elements that are necessary for understanding the methods used.
The first four are common for all methods, while the other two are specific for a method (or a set of methods). For the latter some auxiliary functions or relations are defined later. Therefore we first (Sec. 6) define the common functions, and then (Sec. 8) introduce methods, each of the latter accompanying with the corresponding definitions of AF(·)andSF(·).
6 Objects and functions common for all methods
6.1 Active leaf-criteria
Figure 2: Hierarchy of active criteria
The user may chose to ignore some criteria (cf Sec. 4.1) therefore the set of active leaf-criteria has to be defined for each iteration. This is a trivial operation for analysis without hierarchical structure. If criteria hierarchy is defined (see example in Fig. 1) then criteria at any hierarchy level can be specified as inactive, see example in Fig. 2. In such cases then the set of active-leaf criteria is defined as follows:
•full criteria tree is defined (see Fig. 1).
•Activity of all criteria is defined according the selection of the value of relative criteria importance button (Sec. 4.1).
•active criteria tree is defined by removing from the full criteria tree nodes corresponding to inactive criteria and the branches originating from such nodes.12 Note that activity of criteria imply changes in the impact of the selected relative criteria importance, see Sec. 6.3.
12In the example shown in Fig. 2 only two criteria (marked by white nodes) were selected to be not active. However seven more criteria (marked by gray nodes) become inactive because their parent criteria are inactive.
•The set of active leaf-criteria is composed of leaves of the active criteria tree.
Note: Further on we use the term criteria for the active leaf-criteria, since only such criteria are considered for analysis. Also the number of criteriandenotes the number of active leaf-criteria. The role of intermediate-level active criteria is defined in Sec. 6.3.